# Tagged Questions

Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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I need to calculate the exact value of $1,000,000,000!$ It can't be an approximation. As far as I am aware, the only way to do this would be to calculate it starting from the beginning (1\cdot2\cdot3\... 1answer 40 views ### Do negative binomials imply negative factorials exist? I've seen the following identity: $$\binom{-n}{k} = (-1)^k\binom{n+k-1}{k}$$ So I tried to derive it, assuming negative factorial was a real concept, having it extend down to negative infinity: $$\... 1answer 14 views ### Prove that: {^{n}\mathrm{C}_{k}} = {^{n-1}\mathrm{C}_{k-1}}+{^{n-1}\mathrm{C}_{k}} [duplicate] Question asks to prove: {^{n}\mathrm{C}_{k}} = {^{n-1}\mathrm{C}_{k-1}}+{^{n-1}\mathrm{C}_{k}} My Steps:$$\begin{align*}\frac{(n-1)!}{(n-k-2)!(k-1)!} + \frac{(n-1)!}{(n-k-1)!(k)!} & = \... 1answer 33 views ### Simplifying\frac{^n\mathrm{C}_k}{^n\mathrm{C}_{k-1}}Question asks to simplify: $$\frac{^n\mathrm{C}_k}{^n\mathrm{C}_{k-1}}$$ I have a few steps but not sure if its correct. \begin{align*}\frac{(n)!}{(n-k)!(k)!} \bigg/ \frac{(n)!}{(n-k-1)!(k-1)!}... 2answers 59 views ### simplify factorials: \frac{(k-1)!}{(k+2)!} [duplicate] Question: simplify\frac{(k-1)!}{(k+2)!}$$What I did was:$$\frac{(k - 1)!k!}{(k + 2)! \cdot (k + 1)!}$$This I did following the rule n! = n \times (n - 1)!. can this be simplified ... 2answers 63 views ### Simplifying factorials: \frac{(n-1)!}{(n-2)!} Question: simplify$$\frac{(n-1)!}{(n-2)!}$$What I did was:$$\frac{(n - 1)!}{(n - 2)! \times (n - 3)!}$$This I did following the rule n! = n \times (n - 1)!. But my answer just doesn't look ... 1answer 50 views ### Proof of Stirling's Formula using Trapezoid rule and Wallis Product I need a proof of stirling's formula which uses the riemann's sum and trapezoid approximation to come up with \frac {n!}{(n/e)^n \sqrt n} \rightarrow C where C is derived from Wallis product. ... 1answer 25 views ### Find an explicit map with certain combinatorial properties The following combinatorial problem popped up in a totally uncombinatorial context: Let \mathcal{P} denote the power set of a set and let k \in \mathbb{N}. Is there a map c: \mathcal{P}(\{1,2,\... 1answer 17 views ### Find a map on a power set with certain combinatorial properties The following combinatorial problem popped up in a totally uncombinatorial context: Let \mathcal{P} denote the power set of a set and let k \in \mathbb{N}. Is there a map c: \mathcal{P}(\{1,2,\... 2answers 26 views ### Analytic continuation and how it relates to the gamma function. I am familiar with factorials, and I have read about the gamma function. From what I understand, the gamma function extends the concept of the factorial to complex numbers by nature of being an ... 2answers 80 views ### Finite summation including binomial coefficients and double factorials I came across the following summation:$$ \sum_{k=0}^n\frac{(-1)^k(2k)!!}{(2k+1)!!}\dbinom{n}{k}\,\,\,\,(n\in\mathbb{N}). $$\tbinom{n}{k} are binomial coefficients, n!/k!(n-k)!. Mathematica told ... 1answer 19 views ### recurrence relation - How to determine pattern for an even or odd or different type of factorial Hi I am having trouble on how to solve for the odd terms of recurrence relation in terms of exponential and factorials. How are you able to see a pattern to simplify a non standard factorial. This ... 1answer 56 views ### Is there a more concise expression of this product? In a longer computation, I have stumbled upon the following product, where k,r \in \mathbb{N}_0 are fixed numbers:$$\prod_{0 < i_0<i_1<\dots<i_r\leq k} (i_r-i_{r-1})(i_{r-1}-i_{r-2})\... 1answer 40 views ### notation for factoraling a factorial? (since one cannot do n!!) I was thinking about how to get a number to be larger than graham's number very easily... came up with "factoraling" a factorial. However the notation n!! means something completely different. And I ... 4answers 166 views ### Proof of the summationn!=\sum_{k=0}^n \binom{n}{k}(n-k+1)^n(-1)^k$? I was going through a Number Theory book the other day and found this question. It asked for the proof of the following equation: $$n!=\sum_{k=0}^n \binom{n}{k}(n-k+1)^n(-1)^k$$ I tried hard but ... 2answers 73 views ### Number of zeros at the end of$10^{2}!+ 11^{2}!+12^{2}! \cdots+99^{2}!$How do I find the number of zeros at the end of the Integer $$10^{2}!+ 11^{2}!+12^{2}! \cdots+99^{2}!$$ Answer provided for this question is$24$0answers 23 views ### When is$\frac{2 n f(n)}{n !}$in the order of some fixed power of$n$? I would like to know when$\frac{2 n f(n)}{n !}$is$O (n^b)$where$b$is a constant. Here,$nis a positive integer. My attempt: $$\frac{2 n f(n)}{n !} = \frac{2 n f(n)}{\sqrt{2 \pi n} (\frac{n}{... 2answers 77 views ### Factorial sum estime Prove that:$$\displaystyle \sum_{n=m+1}^\infty \dfrac{1}{n!} \le \dfrac{1}{m\cdot m!}$$I have tried induction on m but it does not work very well. Any suggestion? 5answers 1k views ### Approximation of log(n!) I just finished calculus 1 (derivative and integral) then I take another course on calculus 2. In the video the professor talks about the the series$$\frac{n!}{(\frac{n}{e})^n}$$He shows the ... 1answer 129 views ### A couple of series questions that I just can't figure out (Calc 2) Show that$$ \begin{align} \left(\frac{\pi}{2}\right)^2\left[\int_0^{\pi/2}\cos^{2n}t\ dt-\int_0^{\pi/2}\cos^{2n+2}t\ dt\right]&=\frac{\pi^3}{8}\left[\frac{(2n-1)!!}{(2n)!!}-\frac{(2n+1)!!}{(2n+... 1answer 53 views ### Find all nonnegative integersm$and$n$such that$m!+1=n^2$. [duplicate] This question is inspired by Subgroup of Order$n^2-1$in Symmetric Group$S_n$when$n=5, 11, 71$. Find all nonnegative integers$m$and$n$such that$m!+1=n^2$. We know that$(m,n)=(4,5)$,$(...
$\text{Use the PMI to prove the following for all natural numbers n.}$ $\frac{1}{2!} + \frac{2}{3!} + \cdot \cdot \cdot + \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!}$ So for this question I get ...